Abstract: Using the turbulence differential equation inturbulent mechanics，and Prandtl's hypothesis that shear stress is consrant radially as well as Pranstl's mixing length theory,through accurate mathematical deduction,this paper obtains a new time一averaged velocity distribution formula for the axial turbulent flow in pipe. Because of the existence of the viscous stress ,the formulas are theoretically more rational than Prandtl's turbulent formulas in pipe. Based on the above analyses,this paper also presents the formulas for calculating the turbulence displacement velocity distributions and the interface positions in the pipe of Newtonian fluid.

Key words: Newtonian fluid, tTurbulent flow, Displacement